Pardoux and Peng [1]} first obtained its existence and uniqueness result under the assumption that the generator \(g\)  is linear growth and Lipschitz continuous in \((y,z)\) . Independently, Duffie and Epstein [2]} introduced the notion of recursive utilities in continuous time which is a type of BSDE where \(g\)  is independent of \(z\) . In El Karoui et al. [3]}, a general stochastic control problem with recursive utilities by the solutions to BSDEs, where the generator \(g\)  contains \(z\) , is studied. When the control domain is convex, the global maximum principle reduces to the so-called local maximum principle which is derived by the classical convex variation instead of the spike variation. Peng [4]} first established a local maximum principle for the stochastic recursive optimal control problem, where the state equation of the problem is a controlled forward-backward stochastic differential equation (FBSDE for short). Wu [5]} proved a local maximum principle for the optimal control problem of fully coupled FBSDEs. Wu [6]} and Yong [7]} obtained global maximum principles for optimal control problems of decoupled and fully coupled FBSDEs, respectively.
